The best way to get to San Ignacio is to arrange a transfer service pick you up at the airport and take you. And since you're staying at Parrot's Nest, William's Shuttle would be a good choice. This would also mean you don't have to spend the money for a night in Belize City and the BZ$50 taxi ride. The trip by private vehicle is about 1.5 hours.
The cheapest way is by bus -- about BZ$7 and 2 hours.
Caye Caulker is a good choice.
Do you know about the expess buses from Flores (Guatemala) to Belize City?
For more info about buses & transfer services, click the BelizeBus link on my profile page.
Unless you're arriving very late in the afternoon, I'd try to get to San Ignacio on the day you arrive.
I don’t know your budget, but here are some good LOWER BUDGET places that I know in the City:
Bayview Guesthouse is in a relatively safer neighbourhood on the north side.
http://www.belize-guesthouse-hotel.com/
Freddie’s Guest House, 86 Eve Street, Belize City, private room, shared bath. Cost: about BZ$55.
Three Sisters Guest House, Queen Street, Belize City. Very close to the water taxi terminals and the city center.
And a couple I don’t know as well…
North Front Street Guest House, North Front Street, Belize City. I’ve seen some not so positive comments – bedbugs, noise, etc.
Smokin Balam Guesthouse, 59 North Front Street – not sure if this one’s still open.
There are a couple of hotels on the south side of the city, but they’re in an area of the city I wouldn’t recommend walking around in.
Hotel Mopan, Coningsby Inn.
Isabel’s Guesthouse is very nice, but right in the downtown area of the city – opposite the Swing Bridge.
The Belcove Hotel is also in the downtown area of the city.
More information about these hotels can be found on:
http://www.belizeexplorer.com/
http://www.belizehotels.org/
http://www.travelbelize.org/
Outside the City, closer to the international airport and for more money (and comfort), there’s the D’Nest Inn, 475 Cedar St., Belize City
www.dnestinn.com
There is some other motel style accommodation on the highway between the airport and the City, but without transportation, you're stuck there.
For a bigger budget, there’s the Radisson Fort George and Princess Hotel & Casino in the City, or the Best Western a bit outside the City closer to the airport.